Chapter 13

module Control.Monad (  
    Functor(fmap),  Monad((>>=), (>>), return, fail),  MonadPlus(mzero, mplus),  
    mapM,  mapM_,  forM,  forM_,  sequence,  sequence_,  (=<<),  (>=>),  (<=<),  
    forever,  void,  join,  msum,  filterM,  mapAndUnzipM,  zipWithM,  
    zipWithM_,  foldM,  foldM_,  replicateM,  replicateM_,  guard,  when,  
    unless,  liftM,  liftM2,  liftM3,  liftM4,  liftM5,  ap  
  ) where

The Control.Monad module provides the Functor, Monad and MonadPlus classes, together with some useful operations on monads.

13.1 Functor and monad classes

class Functor f where
The Functor class is used for types that can be mapped over. Instances of Functor should satisfy the following laws:

 fmap id  ==  id  
 fmap (f . g)  ==  fmap f . fmap g

The instances of Functor for lists, Data.Maybe.Maybe and System.IO.IO satisfy these laws.


fmap :: (a -> b) -> f a -> f b

instance Functor []
instance Functor IO
instance Functor Maybe
instance Ix i => Functor (Array i)

class Monad m where
The Monad class defines the basic operations over a monad, a concept from a branch of mathematics known as category theory. From the perspective of a Haskell programmer, however, it is best to think of a monad as an abstract datatype of actions. Haskell’s do expressions provide a convenient syntax for writing monadic expressions.

Minimal complete definition: >>= and return.

Instances of Monad should satisfy the following laws:

 return a >>= k  ==  k a  
 m >>= return  ==  m  
 m >>= (\x -> k x >>= h)  ==  (m >>= k) >>= h

Instances of both Monad and Functor should additionally satisfy the law:

 fmap f xs  ==  xs >>= return . f

The instances of Monad for lists, Data.Maybe.Maybe and System.IO.IO defined in the Prelude satisfy these laws.


(>>=) :: m a -> (a -> m b) -> m b
Sequentially compose two actions, passing any value produced by the first as an argument to the second.

(>>) :: m a -> m b -> m b
Sequentially compose two actions, discarding any value produced by the first, like sequencing operators (such as the semicolon) in imperative languages.

return :: a -> m a
Inject a value into the monadic type.

fail :: String -> m a
Fail with a message. This operation is not part of the mathematical definition of a monad, but is invoked on pattern-match failure in a do expression.

instance Monad []
instance Monad IO
instance Monad Maybe

class Monad m => MonadPlus m where
Monads that also support choice and failure.


mzero :: m a
the identity of mplus. It should also satisfy the equations

 mzero >>= f  =  mzero  
 v >> mzero   =  mzero

mplus :: m a -> m a -> m a
an associative operation

instance MonadPlus []
instance MonadPlus Maybe

13.2 Functions

13.2.1 Naming conventions

The functions in this library use the following naming conventions:

  filter  ::              (a ->   Bool) -> [a] ->   [a]  
  filterM :: (Monad m) => (a -> m Bool) -> [a] -> m [a]

  sequence  :: Monad m => [m a] -> m [a]  
  sequence_ :: Monad m => [m a] -> m ()

  sum  :: Num a       => [a]   -> a  
  msum :: MonadPlus m => [m a] -> m a

13.2.2 Basic Monad functions

mapM :: Monad m => (a -> m b) -> [a] -> m [b]
mapM f is equivalent to sequence . map f.

mapM_ :: Monad m => (a -> m b) -> [a] -> m ()
mapM_ f is equivalent to sequence_ . map f.

forM :: Monad m => [a] -> (a -> m b) -> m [b]
forM is mapM with its arguments flipped

forM_ :: Monad m => [a] -> (a -> m b) -> m ()
forM_ is mapM_ with its arguments flipped

sequence :: Monad m => [m a] -> m [a]
Evaluate each action in the sequence from left to right, and collect the results.

sequence_ :: Monad m => [m a] -> m ()
Evaluate each action in the sequence from left to right, and ignore the results.

(=<<) :: Monad m => (a -> m b) -> m a -> m b
Same as >>=, but with the arguments interchanged.

(>=>) :: Monad m => (a -> m b) -> (b -> m c) -> a -> m c
Left-to-right Kleisli composition of monads.

(<=<) :: Monad m => (b -> m c) -> (a -> m b) -> a -> m c
Right-to-left Kleisli composition of monads. (>=>), with the arguments flipped

forever :: Monad m => m a -> m b
forever act repeats the action infinitely.

void :: Functor f => f a -> f ()
void value discards or ignores the result of evaluation, such as the return value of an IO action.

13.2.3 Generalisations of list functions

join :: Monad m => m (m a) -> m a
The join function is the conventional monad join operator. It is used to remove one level of monadic structure, projecting its bound argument into the outer level.

msum :: MonadPlus m => [m a] -> m a
This generalizes the list-based concat function.

filterM :: Monad m => (a -> m Bool) -> [a] -> m [a]
This generalizes the list-based filter function.

mapAndUnzipM :: Monad m => (a -> m (b, c)) -> [a] -> m ([b], [c])
The mapAndUnzipM function maps its first argument over a list, returning the result as a pair of lists. This function is mainly used with complicated data structures or a state-transforming monad.

zipWithM :: Monad m => (a -> b -> m c) -> [a] -> [b] -> m [c]
The zipWithM function generalizes zipWith to arbitrary monads.

zipWithM_ :: Monad m => (a -> b -> m c) -> [a] -> [b] -> m ()
zipWithM_ is the extension of zipWithM which ignores the final result.

foldM :: Monad m => (a -> b -> m a) -> a -> [b] -> m a
The foldM function is analogous to foldl, except that its result is encapsulated in a monad. Note that foldM works from left-to-right over the list arguments. This could be an issue where (>>) and the ‘folded function’ are not commutative.

       foldM f a1 [x1, x2, ..., xm]


         a2 <- f a1 x1  
         a3 <- f a2 x2  
         f am xm

If right-to-left evaluation is required, the input list should be reversed.

foldM_ :: Monad m => (a -> b -> m a) -> a -> [b] -> m ()
Like foldM, but discards the result.

replicateM :: Monad m => Int -> m a -> m [a]
replicateM n act performs the action n times, gathering the results.

replicateM_ :: Monad m => Int -> m a -> m ()
Like replicateM, but discards the result.

13.2.4 Conditional execution of monadic expressions

guard :: MonadPlus m => Bool -> m ()
guard b is return () if b is True, and mzero if b is False.

when :: Monad m => Bool -> m () -> m ()
Conditional execution of monadic expressions. For example,

       when debug (putStr "Debugging\n")

will output the string Debugging\n if the Boolean value debug is True, and otherwise do nothing.

unless :: Monad m => Bool -> m () -> m ()
The reverse of when.

13.2.5 Monadic lifting operators

liftM :: Monad m => (a1 -> r) -> m a1 -> m r
Promote a function to a monad.

liftM2 :: Monad m => (a1 -> a2 -> r) -> m a1 -> m a2 -> m r
Promote a function to a monad, scanning the monadic arguments from left to right. For example,

    liftM2 (+) [0,1] [0,2] = [0,2,1,3]  
    liftM2 (+) (Just 1) Nothing = Nothing

liftM3 :: Monad m => (a1 -> a2 -> a3 -> r)
                     -> m a1 -> m a2 -> m a3 -> m r
Promote a function to a monad, scanning the monadic arguments from left to right (cf. liftM2).

liftM4 :: Monad m => (a1 -> a2 -> a3 -> a4 -> r)
                     -> m a1 -> m a2 -> m a3 -> m a4 -> m r
Promote a function to a monad, scanning the monadic arguments from left to right (cf. liftM2).

liftM5 :: Monad m => (a1 -> a2 -> a3 -> a4 -> a5 -> r)
                     -> m a1 -> m a2 -> m a3 -> m a4 -> m a5 -> m r
Promote a function to a monad, scanning the monadic arguments from left to right (cf. liftM2).

ap :: Monad m => m (a -> b) -> m a -> m b
In many situations, the liftM operations can be replaced by uses of ap, which promotes function application.

       return f ‘ap‘ x1 ‘ap‘ ... ‘ap‘ xn

is equivalent to

       liftMn f x1 x2 ... xn